$|x^2 + 10x - 4| = 20$
а)
$x^2 + 10x - 4 = 20$
$x^2 + 10x - 4 - 20 = 0$
$x^2 + 10x - 24 = 0$
$D = b^2 - 4ac = 10^2 - 4 * 1 * (-24) = 100 + 96 = 196 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-10 + \sqrt{196}}{2 * 1} = \frac{-10 + 14}{2} = \frac{4}{2} = 2$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-10 - \sqrt{196}}{2 * 1} = \frac{-10 - 14}{2} = \frac{-24}{2} = -12$
б)
$x^2 + 10x - 4 = -20$
$x^2 + 10x - 4 + 20 = 0$
$x^2 + 10x + 16 = 0$
$D = b^2 - 4ac = 10^2 - 4 * 1 * 16 = 100 - 64 = 36 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-10 + \sqrt{36}}{2 * 1} = \frac{-10 + 6}{2} = \frac{-4}{2} = -2$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-10 - \sqrt{36}}{2 * 1} = \frac{-10 - 6}{2} = \frac{-16}{2} = -8$
Ответ: −12; −8; −2; 2.

x|x| + 12x − 45 = 0
а) x ≥ 0
x * x +12x − 45 = 0
$x^2 + 12x - 45 = 0$
$D = b^2 - 4ac = 12^2 - 4 * 1 * (-45) = 144 + 180 = 324 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-12 + \sqrt{324}}{2 * 1} = \frac{-12 + 18}{2} = \frac{6}{2} = 3$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-12 - \sqrt{324}}{2 * 1} = \frac{-12 - 18}{2} = \frac{-30}{2} = -15 < 0$ − не удовлетворяет условию, так как x ≥ 0
б) x < 0
x * (−x) + 12x − 45 = 0
$-x^2 + 12x - 45 = 0$
$D = b^2 - 4ac = 12^2 - 4 * (-1) * (-45) = 144 - 180 = -36 < 0$ − нет корней
Ответ: 3

$\frac{x^3}{|x|} - 14x - 15 = 0$
а) x > 0
$\frac{x^3}{x} - 14x - 15 = 0$
$x^2 - 14x - 15 = 0$
$D = b^2 - 4ac = (-14)^2 - 4 * 1 * (-15) = 196 + 60 = 256 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{14 + \sqrt{256}}{2 * 1} = \frac{14 + 16}{2} = \frac{30}{2} = 15$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{14 - \sqrt{256}}{2 * 1} = \frac{14 - 16}{2} = \frac{-2}{2} = -1 < 0$ − не удовлетворяет условию, так как x ≥ 0
б) x < 0
$\frac{x^3}{-x} - 14x - 15 = 0$
$-x^2 - 14x - 15 = 0$
$D = b^2 - 4ac = (-14)^2 - 4 * (-1) * (-15) = 196 - 60 = 136 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{14 + \sqrt{136}}{2 * (-1)} = \frac{14 + \sqrt{4 * 34}}{-2} = \frac{14 + 2\sqrt{34}}{-2} = \frac{2(7 + \sqrt{34})}{-2} = -7 - \sqrt{34}$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{14 - \sqrt{136}}{2 * (-1)} = \frac{14 - \sqrt{4 * 34}}{-2} = \frac{14 - 2\sqrt{34}}{-2} = \frac{2(7 - \sqrt{34})}{-2} = -7 + \sqrt{34}$
Ответ: $-7 - \sqrt{34}; -7 + \sqrt{34}; 15$.

$x^2 - 8\sqrt{x^2} - 9 = 0$
$x^2 - 8|x| - 9 = 0$
а) x ≥ 0
$x^2 - 8x - 9 = 0$
$D = b^2 - 4ac = (-8)^2 - 4 * 1 * (-9) = 64 + 36 = 100 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{8 + \sqrt{100}}{2 * 1} = \frac{8 + 10}{2} = \frac{18}{2} = 9$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{8 - \sqrt{100}}{2 * 1} = \frac{8 - 10}{2} = \frac{-2}{2} = -1 < 0$ − не удовлетворяет условию, так как x ≥ 0
б) x < 0
$x^2 + 8x - 9 = 0$
$D = b^2 - 4ac = 8^2 - 4 * 1 * (-9) = 64 + 36 = 100 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-8 + \sqrt{100}}{2 * 1} = \frac{-8 + 10}{2} = \frac{2}{2} = 1 > 0$ − не удовлетворяет условию, так как x < 0
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-8 - \sqrt{100}}{2 * 1} = \frac{-8 - 10}{2} = \frac{-18}{2} = -9$
Ответ: −9; 9.